Consider the total Hamiltonian in Eq. (REF ) with the interaction Hamiltonian in Eq. (REF ). Using the following identity [1]}
\(e^{\beta (A+B)} = e^{\beta A}\left[ \mathbb {I} +\int _0^\beta d \lambda e^{-\lambda A} B e^{\lambda (A+B)} \right],\) 
